Titre : | On optimal stochastic control problem of McKean-Vlasov type with some applications via the derivative with respect the law of probability |
Auteurs : | Lina Guenane, Auteur ; Mokhtar Hafayad, Directeur de thèse |
Type de document : | Monographie imprimée |
Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2021 |
Format : | 1 vol. (79 p.) / couv. ill. en coul / 30 cm |
Langues: | Anglais |
Résumé : |
In this thesis, we study the optimal stochastic control for systems governed by McKean- Vlasov stochastic differential equation. of mean-field type. The central theme is the necessary conditions in the form of the Pontryagin’s stochastic maximum of the McKeanVlasov type for optimality with some applications. Recently, the main purpose of this thesis is to derive a set of necessary conditions of optimality, where the differential system is governed by stochastic differential equations of the McKean-Vlasov type. This thesis
is structured around three chapters: In the first chapter, we have presented the different class of stochastic control, such as singular controls, relaxed controls, feedback controls, ergodic controls,..etc. . We briefly write the different the well-known methods of solving a stochastic control problem, which are the dynamic programming method and the Pontryagin maximum principle. In the second chapter, we establish the maximum principle for the optimal control for EDS of McKean-Vlasov type.These results have been proved by Andersson D, Djehiche B, See [7]. In the third chapter, we study singular control problem, where control variable is a pair (u(·), ξ(·)) of measurable A1 × A2−valued, Ft−adapted processes, such that ξ(·) is of bounded variation, non-decreasing continuous on the left with right limits and ξ(0−) = 0. Since dξ(t) may be singular with respect to Lebesgue measure dt, we call ξ(·) : the singular part of the control and the process u(·) : its absolutely continuous part. In this chaptre, we established a new set of necessary conditions of optimal singular control, where the system is governed by stochastic differential equations EDSs. In this work, the control domain is not assumed to be convex (i.e., the control domain is a general action space). The derivatives with respect to measure is applied to establish our new result. The results obtained in Chapter 4 are all new and are the subject of a |
Sommaire : |
Contents
Introduction x 1 Stochastic optimal control problems 14 1.1 Formulation of the control problem . . . . . . . . . . . . . . . . . . . . . . 14 1.1.1 Stochastic process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 Natural fitration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.3 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1.4 Integration by parts formula . . . . . . . . . . . . . . . . . . . . . . 15 1.1.5 Strong formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.6 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Methods to solving optimal control problem . . . . . . . . . . . . . . . . . 18 1.2.1 Dynamic Programming Method . . . . . . . . . . . . . . . . . . . . 18 1.2.2 Pontryagin’s maximum principle . . . . . . . . . . . . . . . . . . . . 23 1.3 Some classes of stochastic controls . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.1 Admissible control . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.2 Optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.3 Near-optimal control . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.4 Singular control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3.5 Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.3.6 Impulsive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.3.7 Ergodic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.3.8 Robust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.3.9 Partial observation control problem . . . . . . . . . . . . . . . . . . 31 1.3.10 Random horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.3.11 Relaxed control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 viCONTENTS vii 2 Maximum Principle for SDE of mean-field type 33 2.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 Necessary conditions for optimality . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1 Taylor Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.2 Adjoint equations and duality . . . . . . . . . . . . . . . . . . . . . 38 2.2.3 Maximum principle for stochastic optimal control . . . . . . . . . . 40 2.3 Sufficient conditions for optimality . . . . . . . . . . . . . . . . . . . . . . 41 3 Optimal singular control problem for general McKean-Vlasov differantial equation 43 3.1 Introduction and brief history . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Novelty in this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Differentiability with respect to measure . . . . . . . . . . . . . . . . . . . 46 3.3.1 Second-order Taylor expansion . . . . . . . . . . . . . . . . . . . . 49 3.4 Formulation of the continuous-singular control problem . . . . . . 51 3.5 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.5.3 Adjoint equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5.4 Necessary conditions of optimal singular control . . . . . . . . . . . 57 Conclusion 68 Bibliographie 71 |
En ligne : | http://thesis.univ-biskra.dz/5484/1/document.pdf |
Disponibilité (1)
Cote | Support | Localisation | Statut |
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TM/112 | Théses de doctorat | bibliothèque sciences exactes | Consultable |